Geodesic
On k-Path Pancyclic Graphs
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 271-281.

Voir la notice de l'article provenant de la source Library of Science

For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G is k-path pancyclic
Keywords: Hamiltonian, panconnected, pancyclic, path Hamiltonian, path pancyclic 
@article{DMGT_2015_35_2_a6,
     author = {Bi, Zhenming and Zhang, Ping},
     title = {On {k-Path} {Pancyclic} {Graphs}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {271--281},
     publisher = {mathdoc},
     volume = {35},
     number = {2},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a6/}
}
TY  - JOUR
AU  - Bi, Zhenming
AU  - Zhang, Ping
TI  - On k-Path Pancyclic Graphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2015
SP  - 271
EP  - 281
VL  - 35
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a6/
LA  - en
ID  - DMGT_2015_35_2_a6
ER  - 
%0 Journal Article
%A Bi, Zhenming
%A Zhang, Ping
%T On k-Path Pancyclic Graphs
%J Discussiones Mathematicae. Graph Theory
%D 2015
%P 271-281
%V 35
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a6/
%G en
%F DMGT_2015_35_2_a6
Bi, Zhenming; Zhang, Ping. On k-Path Pancyclic Graphs. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 271-281. http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a6/

[1] Y. Alavi and J.E. Williamson, Panconnected graphs, Studia Sci. Math. Hungar. 10 (1975) 19-22.

[2] H.C. Chan, J.M. Chang, Y.L. Wang and S.J. Horng, Geodesic-pancyclic graphs, Discrete Appl. Math. 155 (2007) 1971-1978.

[3] G. Chartrand, F. Fujie and P. Zhang, On an extension of an observation of Hamilton, J. Combin. Math. Combin. Comput., to appear.

[4] G. Chartrand, A.M. Hobbs, H.A. Jung, S.F. Kapoor and C.St.J.A. Nash-Williams, The square of a block is Hamiltonian connected, J. Combin. Theory (B) 16 (1974) 290-292. doi:10.1016/0095-8956(74)90075-6

[5] G. Chartrand and S.F. Kapoor, The cube of every connected graph is 1-hamiltonian, J. Res. Nat. Bur. Standards B 73 (1969) 47-48. doi:10.6028/jres.073B.007

[6] G. Chartrand, L. Lesniak and P. Zhang, Graphs & Digraphs: 5th Edition (Chapman & Hall/CRC, Boca Raton, FL, 2010).

[7] G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952) 69-81. doi:10.1112/plms/s3-2.1.69

[8] R.J. Faudree and R.H. Schelp, Path connected graphs, Acta Math. Acad. Sci. Hun- gar. 25 (1974) 313-319. doi:10.1007/BF01886090

[9] H. Fleischner, The square of every two-connected graph is Hamiltonian, J. Combin. Theory (B) 16 (1974) 29-34. doi:10.1016/0095-8956(74)90091-4

[10] C.St.J.A. Nash-Williams, Problem No. 48, in: Theory of Graphs and its Applica- tions, (Academic Press, New York 1968), 367.

[11] O. Ore, Note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55. doi:10.2307/2308928

[12] O. Ore, Hamilton connected graphs, J. Math. Pures Appl. 42 (1963) 21-27.

[13] B. Randerath, I. Schiermeyer, M. Tewes and L. Volkmann, Vertex pancyclic graphs, Discrete Appl. Math. 120 (2002) 219-237. doi:10.1016/S0166-218X(01)00292-X

[14] M. Sekanina, On an ordering of the set of vertices of a connected graph, Publ. Fac. Sci. Univ. Brno 412 (1960) 137-142.

[15] J.E. Williamson, Panconnected graphs II, Period. Math. Hungar. 8 (1977) 105-116. doi:10.1007/BF02018497